Exact divisibility properties of some subsequences of fibonacci numbers

Abstract

For each positive integer n, we consider the following sequence of numbers F(n), F(nF(n)), F(nF(nF(n))), ⋯, where F(n) is the nth Fibonacci number defined in the usual way. Let Gk(n) be the kth term of this sequence. We prove that F(n)k || Gk(n) for all positive integers k and n with n > 3. For the first nontrivial case when n = 3, we prove that F(3)2k-1 || Gk(3) for all positive integers k. We also provide an alternative proof of the divisibility of Gk(n) by F(n)k first proved by two authors of this work. Finally, we give explicit formulas of the quotients obtained from dividing Gk(n) by F(n)k for the cases when k = 2 and k = 3.